[Paper Review] Resonant energy exchange in nonlinear oscillatory chains and Limiting Phase Trajectories: from small to large systems

L. I. Manevich, В. В. Смирнов|ArXiv.org|Mar 30, 2009

Nonlinear Dynamics and Pattern Formation3 references14 citations

TL;DR

This paper introduces the Limiting Phase Trajectory (LPT) as a fundamental framework for analyzing strong energy exchange in weakly coupled nonlinear oscillatory chains, offering an alternative to nonlinear normal modes (NNMs) in systems with complete energy transfer. The LPT enables analytical description of intense intermodal energy exchange, revealing bifurcations and vibro-impact-like dynamics, and explains the emergence of localized excitations (breather-like) in large chains through resonant coupling and separatrix merging.

ABSTRACT

We present an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators and oscillatory chains. The fundamental notion of the limiting phase trajectory (LPT) corresponding to complete energy exchange is introduced. At first we propose a simple analytical description of vibrations of nonlinear oscillators. We show that two dynamical transitions occur in the system. First of them corresponds to the bifurcation of anti-phase vibrations of oscillators. And the second one is caused by coincidence of LPT with separatrix dividing two stable stationary states and leads to qualitative change in both phase and temporal behavior of the LPT. Next problem under consideration relates to intensive intermodal exchange in the periodic nonlinear systems with finite (n>2) number of degrees of freedom. We consider two limiting cases. If the number of particles is not large enough, the energy exchange between nonlinear normal modes in two-dimensional integral manifolds is considered. When the number of the particles increases the energy exchange between neighbor integral manifolds becomes important that leads to formation of the localized excitations resembling the breathers in the one-dimensional continuum media.

Motivation & Objective

• To develop an analytical framework for strong energy exchange in weakly coupled nonlinear oscillators, where traditional nonlinear normal modes (NNMs) fail due to lack of superposition.
• To introduce the Limiting Phase Trajectory (LPT) as a dynamical alternative to NNMs, characterized by complete energy transfer between oscillators.
• To investigate how LPTs evolve with system size, leading to the formation of localized excitations resembling breathers in large periodic chains.
• To analyze the role of frequency resonance, asymmetry, and boundary conditions in triggering energy localization and dynamical transitions.

Proposed method

• Formulates a two-scale asymptotic approach using fast time τ₁ and slow time τ₂ = ετ₁ to derive amplitude equations for complex variables φ₁, φ₂.
• Derives a coupled system of quasilinear differential equations for the slow evolution of mode amplitudes f₁, f₂, capturing energy exchange dynamics.
• Identifies the LPT as the trajectory corresponding to complete energy transfer between two oscillators, defined by phase conditions Δ = 0 (supernormal mode) or Δ = π/2 (elliptical mode).
• Analyzes bifurcations via separatrix formation, particularly when the LPT coincides with the separatrix dividing stable stationary states.
• Applies the LPT framework to periodic chains with finite N > 2, examining intermodal coupling in 2D integral manifolds and transition to 3D dynamics as N increases.
• Uses numerical simulations to validate the emergence of localized excitations and chaotic breathers via separatrix merging and frequency detuning.

Experimental results

Research questions

• RQ1 How does the Limiting Phase Trajectory (LPT) describe complete energy exchange in weakly coupled nonlinear oscillators, and how does it differ from nonlinear normal modes (NNMs)?
• RQ2 What dynamical transitions occur in the system as parameters such as coupling strength and asymmetry vary, particularly near bifurcation points?
• RQ3 How does the LPT evolve in periodic chains with increasing number of degrees of freedom, and what leads to the formation of localized excitations resembling breathers?
• RQ4 What role does quasi-resonance between high-frequency modes (e.g., π-mode) and nearby modes play in energy localization and the onset of chaotic behavior?
• RQ5 How do boundary conditions and system length influence the onset of energy confinement and the merging of separatrix and LPT trajectories?

Key findings

• The LPT provides a complete analytical description of strong energy exchange in two weakly coupled nonlinear oscillators, with temporal amplitude profiles resembling vibro-impact vibrations when the LPT coincides with the separatrix.
• Two dynamical transitions occur: (1) a bifurcation of anti-phase vibrations, and (2) a critical transition when the LPT merges with the separatrix, leading to qualitative changes in phase and temporal dynamics.
• For finite N > 2, resonant coupling between degenerate NNMs in 2D integral manifolds leads to slow energy transfer, while increasing N enables coupling between adjacent manifolds, promoting localized excitation formation.
• Energy localization occurs when the separatrix of the π-mode merges with the LPT at a critical energy threshold E = π²/(3βN), independent of chain length, leading to confinement of energy in a bounded region.
• The critical occupation number for localization is X_c = 16π²/(27β), above which the system exhibits a new LPT that separates closed and transit-time trajectories, marking the onset of chaotic breather dynamics.
• Asymmetry in the system suppresses complete energy transfer: complete exchange ceases when asymmetry decreases below a threshold, restoring stability of the supernormal mode while destabilizing the elliptical mode.